Degree of a Vertex (Advanced Applications and Extensions)

In digraphs, each vertex v has two measures:

in-degree deg⁻(v): edges entering v

out-degree deg⁺(v): edges leaving v

Total degree = deg⁻(v) + deg⁺(v).

The correct statement is B because a directed edge A → B leaves A and therefore contributes +1 to A’s out-degree while entering B and contributing +1 to B’s in-degree, illustrating the rule that each directed edge increases one vertex’s out-degree and another vertex’s in-degree, with total in-degrees and total out-degrees both equaling the number of edges.

The correct answer is 20 because in a digraph every directed edge contributes exactly one unit to some vertex’s in-degree, so the sum of all in-degrees equals |E| = 20, and by symmetry the total out-degree sum is also 20.

A regular digraph with n vertices and constant out-degree r has n × r edges because each of the n vertices emits exactly r outgoing edges, and since each edge is counted exactly once at its tail, multiplying n by r gives the total number of directed edges in the graph.

The correct implication is that the junction accumulates 5 cars per hour because the total in-flow of 100 cars/hour exceeds the out-flow of 95 cars/hour, meaning a net +5 cars/hour build up at that vertex, which reflects the flow-balance principle used in network modeling where excess in-flow indicates accumulation.

Statements B and C are true because summing the degrees 2 + 2 + 3 + 3 + 4 + 4 yields 18, and dividing by 2 gives 9 edges according to the Handshaking Lemma, while statement A is false since a regular graph requires all degrees to be equal and statement D is false because there are actually four even-degree vertices (2, 2, 4, 4), not three.

The weighted degree of v is 20 because in an undirected weighted graph the weighted degree is defined as the sum of the weights of all edges incident to v, so adding 4, 6, and 10 gives 20, which represents the total cost or capacity associated with that vertex.

Relation A must always hold because each directed edge increases exactly one vertex’s out-degree and exactly one vertex’s in-degree, meaning Σ deg⁺(v) = Σ deg⁻(v) = |E|, and none of the other listed relations consistently hold for all digraphs.

True, because a loop in a directed or undirected setting simultaneously leaves and enters the same vertex, contributing +1 to in-degree and +1 to out-degree, giving a total degree contribution of 2.

True, because a balanced digraph is defined by the condition deg⁺(v) = deg⁻(v) for every vertex v, representing systems with no accumulation or deficit of flow at any node.

False, because in a simple digraph a vertex cannot have an edge to itself, so the maximum possible out-degree is n − 1 rather than n.

True, because a vertex with out-degree 0 emits no edges and only receives incoming edges, which is exactly the definition of a sink in directed graph terminology.

False, because having all vertices of even degree does not guarantee connectivity—disconnected structures like several separate cycles can each have all-even degrees yet still form a disconnected graph.

True, because the sum of degrees in any undirected graph equals 2|E|, which is always an even number since twice any integer is even.

In a directed graph the total number of edges equals both the sum of all out-degrees and the sum of all in-degrees because each edge contributes exactly one unit to the out-degree of its tail and one unit to the in-degree of its head, ensuring Σ deg⁺ = Σ deg⁻ = |E|.

A vertex with in-degree 0 and out-degree greater than 0 is called a source because it emits edges but receives none, functioning as a starting point or origin in flow, dependency, or causal graphs.

If every vertex in an undirected graph has degree 2, the graph consists of one or more cycles because each vertex lies on exactly two edges, forcing the edges to connect in closed loops with no branching or endpoints.

The maximum number of edges in a simple undirected graph with n vertices is n(n − 1)/2 because each vertex can connect to n − 1 distinct others but undirected edges are counted twice in that total, so dividing by 2 yields the true number of unique edges.

If vertex A has degree 5 in a 6-vertex graph, it is adjacent to 5 other vertices because degree counts the number of distinct neighbors in a simple graph, and the only vertex it cannot connect to is itself.

A vertex with both in-degree and out-degree equal to 0 is called an isolated vertex because it neither sends nor receives edges, making it disconnected from the rest of the digraph and giving it total degree 0.